Theorem caucvgprlemladdfu 7639

Description: Lemma for caucvgpr 7644. Adding S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
caucvgprlemladd.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
caucvgprlemladdfu	|- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )

Distinct variable groups:   A, j   j, F, u, l   n, F, k   k, L, j   S, l, u, j   j, k

Allowed substitution hints:   ph( u, j, k, n, l)   A( u, k, n, l)   S( k, n)   L( u, n, l)

Proof of Theorem caucvgprlemladdfu

Dummy variables m r s t v w z f g h x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . . . . 7 |- ( ph -> F : N. --> Q. )
2		caucvgpr.cau	. . . . . . 7 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
3		caucvgpr.bnd	. . . . . . 7 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		caucvgpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
5	1, 2, 3, 4	caucvgprlemcl 7638	. . . . . 6 |- ( ph -> L e. P. )
6		caucvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7509	. . . . . . 7 |- ( S e. Q. -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
8	6, 7	syl 14	. . . . . 6 |- ( ph -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
9		df-iplp 7430	. . . . . . 7 |- +P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
10		addclnq 7337	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvu 7475	. . . . . 6 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
12	5, 8, 11	syl2anc 409	. . . . 5 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
13	12	biimpa 294	. . . 4 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) )
14		breq2 3993	. . . . . . . . . . . . . . . 16 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
15	14	rexbidv 2471	. . . . . . . . . . . . . . 15 |- ( u = s -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
16	4	fveq2i 5499	. . . . . . . . . . . . . . . 16 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
17		nqex 7325	. . . . . . . . . . . . . . . . . 18 |- Q. e. _V
18	17	rabex 4133	. . . . . . . . . . . . . . . . 17 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
19	17	rabex 4133	. . . . . . . . . . . . . . . . 17 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
20	18, 19	op2nd 6126	. . . . . . . . . . . . . . . 16 |- ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
21	16, 20	eqtri 2191	. . . . . . . . . . . . . . 15 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
22	15, 21	elrab2 2889	. . . . . . . . . . . . . 14 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
23	22	biimpi 119	. . . . . . . . . . . . 13 |- ( s e. ( 2nd ` L ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
24	23	adantr 274	. . . . . . . . . . . 12 |- ( ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
25	24	adantl 275	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
26	25	adantr 274	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
27	26	simpld 111	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> s e. Q. )
28		vex 2733	. . . . . . . . . . . . . 14 |- t e. _V
29		breq2 3993	. . . . . . . . . . . . . 14 |- ( u = t -> ( S <Q u <-> S <Q t ) )
30		ltnqex 7511	. . . . . . . . . . . . . . 15 |- { l | l <Q S } e. _V
31		gtnqex 7512	. . . . . . . . . . . . . . 15 |- { u | S <Q u } e. _V
32	30, 31	op2nd 6126	. . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) = { u | S <Q u }
33	28, 29, 32	elab2 2878	. . . . . . . . . . . . 13 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> S <Q t )
34		ltrelnq 7327	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
35	34	brel 4663	. . . . . . . . . . . . 13 |- ( S <Q t -> ( S e. Q. /\ t e. Q. ) )
36	33, 35	sylbi 120	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> ( S e. Q. /\ t e. Q. ) )
37	36	simprd 113	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t e. Q. )
38	37	ad2antll 488	. . . . . . . . . 10 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> t e. Q. )
39	38	adantr 274	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> t e. Q. )
40		addclnq 7337	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
41	27, 39, 40	syl2anc 409	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s +Q t ) e. Q. )
42		eleq1 2233	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
43	42	adantl 275	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	41, 43	mpbird 166	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. Q. )
45	26	simprd 113	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
46		fveq2 5496	. . . . . . . . . . . . 13 |- ( j = m -> ( F ` j ) = ( F ` m ) )
47		opeq1 3765	. . . . . . . . . . . . . . 15 |- ( j = m -> <. j , 1o >. = <. m , 1o >. )
48	47	eceq1d 6549	. . . . . . . . . . . . . 14 |- ( j = m -> [ <. j , 1o >. ] ~Q = [ <. m , 1o >. ] ~Q )
49	48	fveq2d 5500	. . . . . . . . . . . . 13 |- ( j = m -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. m , 1o >. ] ~Q ) )
50	46, 49	oveq12d 5871	. . . . . . . . . . . 12 |- ( j = m -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
51	50	breq1d 3999	. . . . . . . . . . 11 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s <-> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) )
52	51	cbvrexv 2697	. . . . . . . . . 10 |- ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s <-> E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
53	45, 52	sylib 121	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
54	33	biimpi 119	. . . . . . . . . . . . . . . . 17 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> S <Q t )
55	54	ad2antll 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> S <Q t )
56	55	adantr 274	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> S <Q t )
57	56	ad2antrr 485	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> S <Q t )
58	6	ad5antr 493	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> S e. Q. )
59	39	ad2antrr 485	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> t e. Q. )
60	1	ad5antr 493	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> F : N. --> Q. )
61		simplr 525	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> m e. N. )
62	60, 61	ffvelrnd 5632	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( F ` m ) e. Q. )
63		nnnq 7384	. . . . . . . . . . . . . . . . 17 |- ( m e. N. -> [ <. m , 1o >. ] ~Q e. Q. )
64		recclnq 7354	. . . . . . . . . . . . . . . . 17 |- ( [ <. m , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
65	61, 63, 64	3syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
66		addclnq 7337	. . . . . . . . . . . . . . . 16 |- ( ( ( F ` m ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
67	62, 65, 66	syl2anc 409	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
68		ltanqg 7362	. . . . . . . . . . . . . . 15 |- ( ( S e. Q. /\ t e. Q. /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. ) -> ( S <Q t <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) ) )
69	58, 59, 67, 68	syl3anc 1233	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( S <Q t <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) ) )
70	57, 69	mpbid 146	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) )
71		simpr 109	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
72		ltanqg 7362	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( z <Q w <-> ( v +Q z ) <Q ( v +Q w ) ) )
73	72	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) /\ ( z e. Q. /\ w e. Q. /\ v e. Q. ) ) -> ( z <Q w <-> ( v +Q z ) <Q ( v +Q w ) ) )
74	27	ad2antrr 485	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> s e. Q. )
75		addcomnqg 7343	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) = ( w +Q z ) )
76	75	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) /\ ( z e. Q. /\ w e. Q. ) ) -> ( z +Q w ) = ( w +Q z ) )
77	73, 67, 74, 59, 76	caovord2d 6022	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) ) )
78	71, 77	mpbid 146	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) )
79		ltsonq 7360	. . . . . . . . . . . . . 14 |- <Q Or Q.
80	79, 34	sotri 5006	. . . . . . . . . . . . 13 |- ( ( ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) /\ ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( s +Q t ) )
81	70, 78, 80	syl2anc 409	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( s +Q t ) )
82		simpllr 529	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> r = ( s +Q t ) )
83	81, 82	breqtrrd 4017	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
84	83	ex 114	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
85	84	reximdva 2572	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s -> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
86	53, 85	mpd 13	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
87	50	oveq1d 5868	. . . . . . . . . 10 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) = ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) )
88	87	breq1d 3999	. . . . . . . . 9 |- ( j = m -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
89	88	cbvrexv 2697	. . . . . . . 8 |- ( E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r <-> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
90	86, 89	sylibr 133	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r )
91		breq2 3993	. . . . . . . . 9 |- ( u = r -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u <-> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
92	91	rexbidv 2471	. . . . . . . 8 |- ( u = r -> ( E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u <-> E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
93	92	elrab 2886	. . . . . . 7 |- ( r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } <-> ( r e. Q. /\ E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
94	44, 90, 93	sylanbrc 415	. . . . . 6 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
95	94	ex 114	. . . . 5 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( r = ( s +Q t ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
96	95	rexlimdvva 2595	. . . 4 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
97	13, 96	mpd 13	. . 3 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
98	97	ex 114	. 2 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
99	98	ssrdv 3153	1 |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   C_ wss 3121   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   2ndc2nd 6118   1oc1o 6388   [cec 6511   N.cnpi 7234   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247   P.cnp 7253   +P. cpp 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iplp 7430

This theorem is referenced by:  caucvgprlemladdrl  7640